Let me label the diagram you posted with the terminology I am using. We call the distance from the point "Camera" to the point "Subject" the subject distance, which we might label S. The distance from "Subject" to "Lights" is the subject-to-background distance, or B. We will use the traditional F for for focal length and N for f-number. Finally, not directly shown in your diagram is the subject magnification M, which is the ratio of the size of the subject on the image plane, to the size of the subject in the object plane. In other words, M is simply the size of the subject in focus on the sensor divided by the size of the same subject in real life.
When you say that you want to maintain the same "frame of view," what you are actually meaning is that you wish for M to remain the same. For a pair of different focal lengths as you have drawn in your diagram, what this requires is a change in S, which you have also drawn. You have also shown that because F relates to angle of view, changing S also changes the relative size of the blur circle at S+B.
But as you have correctly surmised, by stopping down in the lower diagram, you are able to decrease the size of the blur circle to match those in the upper diagram. Alternatively, you have also correctly inferred that, if the f-numbers of the upper and lower diagrams are fixed at, say, f/1.8 and f/4, then there is a relationship between the focal lengths that would result in the same degree of background blur.
However, what my previous posts have explained is that this relationship would only hold true for a SPECIFIC subject distance S and a SPECIFIC subject-to-background distance B. If you changed either of these, then the focal length required in the lower diagram to match the degree of blur in the upper diagram will also change. Even worse, there are certain combinations of S and B for which NO focal length in the lower diagram will give you the same degree of blur.